Document FR 2 815 610 describes a fastener device comprising a ring and a hoop. The ring receives the pump or the valve in stationary manner, and includes a peripheral skirt for coming into engagement with the outside of the neck of the reservoir that forms an outer annular reinforcement that defines a bottom rim that serves as a fastener zone for the skirt. To do this, the skirt includes a plurality of tabs that each define a deformation zone that is adapted to deform radially inwards below the bottom rim of the neck. To achieve the deformation, it is necessary to exert axial pressure on the bottom free ends of the tabs. In practice, the bottom ends of the tabs come to bear against a shoulder of the reservoir, situated at the base of the neck. In other words, the skirt bears against the shoulder, and by exerting axial pressure on the ring, the deformation zone deforms radially inwards below the rim of the neck. In order to guarantee that the deformation zones of the tabs do deform inwards, the hoop is used that surrounds the skirt of the ring. In other words, the hoop prevents the tabs of the skirt from deforming outwards. However, the deformation zone of the tabs include two hinged connection pieces so that the bottom sections of the tabs pivot inwards while the skirt is deforming. It can thus be said that the deformation zone extends over the entire bottom portion of the tabs. Thus, the skirt comes into contact with the hoop only at certain points at its bottom end. Unfortunately, such point contact does not suffice to hold the ring firmly inside the hoop, most particularly at its bottom end. Tests have made it possible to see this weakness in the fastener device: by pulling on the hoop, it is possible to remove the fastener device from the neck, enabling the bottom section of the ring to return substantially to its non-deformed original position. It was quickly concluded that that retention fault results from the bottom section of the ring being poorly fastened inside the hoop. It is necessary to keep firmly in mind that such a fastener device is for ensuring that the dispenser member is fastened not only in leaktight manner, but also in permanent manner, on or in the neck of the reservoir.